SPACE-TIME DECOMPOSITION FOR ANALYZING FLUCTUATIONS IN FRANCIS TURBINES

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1 6 th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, September 9-11, 2015, Ljubljana, Slovenia SPACE-TIME DECOMPOSITION FOR ANALYZING FLUCTUATIONS IN FRANCIS TURBINES Josselin GUILLOZET* Alstom Hydro France, Grenoble, France Alexandre DUPARCHY Alstom Hydro France, Grenoble, France Thomas DE COLOMBEL Alstom Hydro France, Grenoble, France ABSTRACT The spatial harmonic decomposition (SHD) is a new analysis method that allows a broader perspective into the instabilities occurring in Francis turbines at off-design points. From a set of pressure signals distributed in space, this method offers an expansion of the pressure fluctuations into relevant patterns. It is a more robust and complete method than existing processes, that already distinguish between synchronous and asynchronous fluctuations. Such methods and physical analyses allow a better understanding of Francis turbines behavior at part load and deep part load. Indeed Francis turbines are increasingly required to operate at off-design points to integrate intermittent renewable energy source into the electrical grid. In such conditions, the flow may be unstable and lead to strong fluctuations. A way to assess associated risks has long been to measure wall pressure fluctuations at different locations of the reduced scale turbine. However rough peak-to-peak levels of these pulsations do not tell much about the actual solicitations for the runner and which part of the fluctuation may be transposable to full scale machine. After a short mathematical description of SHD, this paper will discuss its application to model Francis turbines. KEYWORDS Pressure Pulsations, Part Load, Deep Part Load, Francis Turbine, Stability, Vortex Rope, Hydroacoustic Resonance 1. INTRODUCTION Today, the stability of a turbine is mainly judged on reduced scale model by analyzing peakto-peak value of pressure pulsations measurements. The sensors are commonly located at the spiral case inlet, in the vaneless area and in the draft tube. For a given net head, these peak-topeak values are then expressed as a percentage of the net head against the power output and supposed to transpose as such to the full scale machine. Such a rough peak-to-peak analysis illustrated on Fig. 1 suffers at least two major drawbacks. * Corresponding author: Alstom Hydro France, 82 Av. Léon Blum - BP 75,38041 Grenoble Cedex 9, France, phone: , josselin-a.guillozet@power.alstom.com

2 Fig. 1 Traditionnal specifications in terms of peak-to-peak Firstly, it does not tell whether the pulsations will actually decrease the lifetime of components or induce power fluctuations. For the same peak-to-peak value of pressure pulsations, a synchronous instability (as defined in [3]) and a rotating pattern may have very different impacts. The left chart of Fig. 2 shows that synchronous resonances are often not visible in the runner strain fluctuations contrary to the rotating vortex rope. On the other side, synchronous instabilities usually generate more power fluctuations than rotating patterns as illustrated on the right chart of Fig. 2. Fig. 2 Left, strain fluctuations in a reduced scale runner and cone pressure pulsations. Right, power and pressure pulsations on a full scale turbine before enabling mitigation device from [6]. Secondly, peak-to-peak analysis does not make any difference between phenomena that transpose more or less easily to the full scale turbine. For example, hydro-acoustic resonances may depend on the distance between the upstream and downstream tanks, which is not always in homology from the reduced scale to the full scale turbine. Furthermore, resonances are very sensitive to the cavitation volume and hence to both Thoma and Froude similitude. Unfortunately, Froude similitude is not always compatible at reduced scale with range of Reynolds numbers required for the transposition of losses to full scale turbine. The SHD solves these problems by making visible the space shape of pressure pulsations on any cross section. Indeed, thanks to this method, synchronous pulsations and various rotating pattern are separated. Once each cross section is analyzed, SHD patterns may be combined to exhibit spatial shape in the third dimension: from the inlet to the outlet of the turbine. After a short review of the mathematical properties of the SHD first developed in [1], this paper will discuss its application to common pressure pulsation measurements on Francis turbines.

3 2. SHD MATHEMATICAL PROPERTIES The details of the mathematical development for the SHD may be found in [1]. This paragraph gives the main properties of this method. The pressure pp is considered on a circle perpendicular to and centered on the runner axis. Such a circle may be defined on a draft tube cross section or on a ring above or below the vaneless area. On this circle, the pressure is only function of time tt and of the azimuth αα. The periodicity of pp(αα, tt) in variable αα allows an expansion into a Fourier series. Then, on top of this first expansion for the spatial variable, a more common Fourier transform is applied for the time tt. This double Fourier expansion leads to following final decomposition: kk=+ pp(αα, tt) = AA 0,ωω cos (ωωωω + φφ 0,ωω ) dddd + AA kk,ωω cos(kkkk ωωωω + φφ kk,ωω ) dddd ωω=0 kk= 1 kk=+1 ωω=0 + AA kk,ωω cos(kkkk ωωωω + φφ kk,ωω ) dddd (1) kk= ωω=0 Each of these terms has the following physical interpretation: PP 00,ωω (tt) = AA 00,ωω cccccc (ωωωω + φφ 00,ωω ) is a synchronous pattern. If sensors are equally distributed, it is the space average of all the sensors. PP kk,ωω (tt) = AA kk,ωω cos(kkkk ωωωω + φφ ωω,tt ) with kk > 0 is a rotating pattern with k nodal diameter(-s). The pattern rotates in the direction of αα > 0 with angular velocity αα = ωω kk. PP kk,ωω (tt) = AA kk,ωω cos(kkkk ωωωω + φφ ωω,tt ) with kk < 0 is a pattern with k nodal diameter(s) rotating in the opposite direction. Some of these patterns are pictured on Fig. 3 together with corresponding time series on four equally distributed sensors. Fig. 3 Illustration of the two most common PP kk patterns to be found in a Francis turbine.corresponding time series from 4 sensors on the right of each pattern. In practice, the time is acquired at a limited sampling frequency ff ss. Likewise, regarding the space variable αα, a limited number NN αα of sensors are installed. According to Nyquist- Shannon theorem, a sampling frequency ff ss gives only access to phenomena up to a frequency ff max = ff ss /2. Higher frequency will experience aliasing, that is, they will be mistaken for lower frequency. Similarly, NN αα gives only access to patterns up to kk mmmmmm = ±NN αα /2. In the next paragraph, every time SHD is applied, 3 or 5 sensors have been used, giving access to patterns having up to 1 or 2 nodal diameter. Pattern with a number of nodal diameters kk higher than NN αα /2 will be seen as a pattern with the parameter:

4 kk mmmmmmmmmmmmmmmm = kk rrrrrrrr modulo NN aaaaaahaa (2) For example, if only NN αα =5 sensors are available, the passage of 15 blades in the vaneless area expected to be mainly seen on PP 15,rrrrrrrr will be captured by PP 0,mmmmmmmmmmmmmmmm, because 15 = It should be noted that not only SHD but all methods exploring the spatial shape of fluctuations will experience aliasing but SHD provides the mathematical framework to explain it. To quantify the relative contribution of each pattern to the overall signal, Parseval s theorem may be invoked on signal powers. Total signal power of the fluctuations on a section may be defined as: ππ TT PPPPPPPPPP(pp) = 1 2ππ 1 TT pp2 (αα, tt)dddd dddd = ππ tt=0 Total Signal Power kk=+ kk= ωω=0 1 2 AA 2 kk,ωω Contribution of one single pattern dddd (3) After spatial discretization with NN αα sensors and introduction of Root-Mean-Square (RMS) over time or frequency: NN αα PPPPPPPPPP(pp) = 1 RRRRSS 2 (pp NN ii ) αα ii=1 Total Signal Power kk=+kk mmmmmm = RRRRSS 2 (PP kk ) kk= kk mmmmmm Contribution of one single pattern It allows one to answer the question: what is the contribution of each pattern to the overall signal? 3. APPLICATION TO A FRANCIS TURBINE The investigated case is a reduced scale model of a medium head Francis turbine. The runner has ZZ bb = 15 blades. Two load variations at a constant net head and constant runner speed ff 0 have been performed from speed-no-load to very high load for two different Thoma numbers σσ hiiiih and σσ llllll. One sensor is located at the spiral case inlet. Three sensors are equally distributed on a circle at the top of the vaneless area between the wicket gate and the runner inlet. Five sensors are located in the cone below the runner outlet. σσ hiiiih is a high Thoma number which gives little cavitation and which is not representative of a full scale machine. However, operating points at σσ hiiiih may be compared with single-phase flow numerical calculation. Load variation with σσ llllll is more representative of a real plant downstream level and exhibits resonances and instabilities that are strongly dependent on the cavitation volume in the draft tube. On Fig. 4, the total signal power is plotted for both the vaneless area and the cone against the discharge QQ nondimensionalized by the optimum discharge QQ oooooo. Some time series and spectra are also given for a selection of four operating points on Fig. 5. In-phase patterns PP 0 are pictured on Fig. 6. They highlight: 1. Small resonances at partial load (e.g. OP3): two resonant frequencies can be seen at ff rr1 = 0.4ff 0 (very close to the vortex rope second harmonic) and ff rr2 = 1.2ff 0. They are present everywhere in the turbine. Such resonances were investigated recently in [4]. They are natural hydro-acoustic modes whose frequency and spatial shape depend mainly on the turbine length and on the cavitation volume. Some of these modes are probably excited by the vortex rope interacting with the draft tube elbow. (4)

5 2. Small high load instabilities (OP4): One of them is here visible at frequency ff HHHHHH = 0.6ff 0. This kind of self-excited oscillation is due to the interaction between the runner and the cavitation volume in the draft tube. It is described for example in [5]. Like the resonances, it oscillates in phase everywhere in the turbine (see Fig. 5). 3. A small PP 0 contribution by the helical vortex at partial load due to the elbow as identified in [2]. Fig. 4 Total signal power for draft tube and vaneless area sections. Unfiltered data for the spiral case inlet is also pictured. Four interesting operating points are labeled for discussion. Fig. 5 Time series and corresponding FFT spectrum for interesting operating points labeled on Fig. 4. Amplitude scales are conserved from one OP to the other. At deep partial load (below 0.6Qopt and before speed-no-load), part of the power of PP 0 may be due to the aliasing of rotating patterns with a number of nodal diameter multiple of 5. Such patterns are probable, for example in the draft tube where five vortices may appear. More sensors on the same section would have allowed a better identification.

6 Fig. 6 Signal power of synchronous patterns from SHD for draft tube and vaneless area sections. Unfiltered data for the spiral case inlet is also pictured. Rotating patterns PP kk are plotted on Fig. 7. They are due to vortices rotating in the draft tube or to oscillating flow located in the vaneless area. They are related to a convective peripheral velocity VV uu at radius RR that drives their angular velocity αα VV uu /RR. Fig. 7 Signal power of rotating patterns from SHD for draft tube and vaneless area sections. Unfiltered data for the spiral case inlet is also pictured. Between 0.6Qopt and 0.9Qopt, the well-known helical vortex rope present on every Francis draft tube is captured mainly by a PP 1 pattern plotted in red line. Its main frequency is ff VVVV = 0.25ff 0. When the vortex gets closer to the cone wall for lower discharge, a second harmonic 2ff VVVV appears that is captured by a small PP 2 pattern. At deep partial load, the situation is less obvious but some interpretations can be made. Below 0.55Qopt, inter-blade vortices (IBV) are the main cavitating phenomenon visible inside the runner. In the stationary frame, their expected frequency signature might range from 0.2Z b f 0 = 3ff 0 (15 vortices convected like the partial load vortex rope) to Z b f 0 (15 vortices precessing at runner speed). Fluctuations above 5f 0 are actually recorded mainly by pressure sensors in the vaneless area below 0.4Qopt (see Fig. 5). In this particular turbine, they may be good candidates to be signatures of IBV. In the cone, mainly frequencies below 3ff 0 can be observed. These frequencies may be interpreted in the continuity of the range Qopt as 1 to 5 rotating vortices similar to the helical vortex rope. Nevertheless these vortices are not

7 stable and the number of them may change over time at the same operating point. Timefrequency or time-shd analysis should support this affirmation. Contrary to the helical vortex rope, draft tube vortices at deep part load seem to only have a local impact confined to the draft tube itself. Finally, pressure sensors in the cone are of little use to investigate what happens in the runner at deep part load. Vaneless area measurements seem to be a bit more meaningful to assess risks due to IBV. Nevertheless, fluctuations may have been harmful for the runner before their detection in the vaneless area. Therefore, only on-board measurements may decide about the actual risk for the runner. Fig. 8 Impact of Thoma number on SHD patterns for two Thoma numbers Another aspect that supports the separation of synchronous from rotating patterns is their respective sensitivity to the Thoma number. This property is illustrated on Fig. 8 with dotted curves for the lower Thoma number. Rotating patterns related mainly to flow convection are little impacted by a change in the counter-pressure (red and orange curves) on a broad operating range. On the contrary, partial load resonances and high load instabilities which strongly depend on the cavitation volume, change much more with the Thoma number (blue curves). 4. CONCLUSION Spatial Harmonic Decomposition was applied to pressure pulsations on a Francis turbine. Thanks to this space-time decomposition, unstable hydraulic phenomena are easily identified and their contribution to the overall fluctuations is well quantified. It is a powerful alternative to peak-to-peak analysis and project specifications would be more suitably express in terms of SHD patterns power. Once phenomena have been identified on a reduced or an existing full scale machine, they can be addressed with the appropriate mitigation measure. The next step would consist in applying the SHD to a network of pressure sensors inside the runner to have a better insight of the phenomena at deep part load. Indeed, common pressure sensors in the vaneless area and in the draft tube fail to deliver a clear view of what happens at such operating points.

8 5. REFERENCES [1] Duparchy A, Guillozet J, de Colombel T and Bornard L: Spatial Harmonic Decomposition as a tool for unsteady flow phenomena analysis 27th IAHR Symposium on Hydraulic Machinery and Systems (IAHR) [2] Nishi M, Matsunaga S, Kubota S and Senoo Y: Surging Characteristics of conical and elbow-type draft tubes. IAHR Section Hydraulic Machinery, Equipment and Cavitation, 12th Symp. (Stirling, UK) [3] Doerfler P K: System dynamics of the Francis turbine half-load surge. IAHR Section Hydraulic Machinery, Equipment and Cavitation, 11th Symp. (Amsterdam, Netherlands) [4] Favrel A, Landry C, Mueller A, Yamamoto K and Avellan F. Hydro-acoustic resonance behavior in presence of a precessing vortex rope: observation of a lock-in phenomenon at part load Francis turbine operation. 27th IAHR Symposium on Hydraulic Machinery and Systems (Montreal, Canada) [5] Mueller A, Favrel A, Landry C, Yamamoto K and Avellan F: On the physical mechanisms governing self-excited pressure surge in Francis turbines. 27th IAHR Symposium on Hydraulic Machinery and Systems (Montreal, Canada) [6] Flores E, Bornard L, Tomas L, et al. : Design of large Francis turbine using optimal methods. 26th IAHR Symposium on Hydraulic Machinery and Systems (Beijing, China) NOMENCLATURE pp (Pa) Pressure kk (-) Number of diametric node αα (rad) Spatial azimuth PP kk,ωω (Pa) Single SHD pattern AA kk,ωω (Pa) Pressure signal amplitude of a SHD pattern φφ kk,ωω (rad) Phase of a SHD pattern ωω (rad/s) Time pulsation ωω 0 or f 0 (rad/s or Hz) Runner rotation speed NN αα (-) Number of sensors on a ZZ bb (-) Runner blade number cross-section QQ (l/s) Flow Rate σσ (-) Thoma number αα (rad/s) Angular rotation speed VV uu (m/s) Peripheral velocity RR (m) Radius